Explore the concept of Topological Complexity (TC) and its application to the motion planning problem in robotics in this comprehensive lecture. Delve into Michael Farber's groundbreaking model, which has garnered significant attention due to its rich algebraic topology properties. Review the basic constructions and properties of TC before examining recent developments and research trends in this fruitful area. Learn about motion planners, optical motion planners, motion planner instability, and theoretical aspects of motion planning. Investigate general properties, homotopy, topology, Snyderman category, and the relationship between topological complexity and Cartesian products. Discover intriguing questions surrounding topological complexity, including whether it is an equality and the concept of Ecuadorian topological complexity.